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Let $\mathcal{C}$ be a collection of (isomorphism classes of) finite groups with the following properties: If $G\in\mathcal{C}$ and $H$ is a homomorphic image of $G$, then $H\in\mathcal{C}$ If $G\in\ …

The broad theme that underlies this question is: to what extent can the study of finite groups be reduced to the study of $p$-groups? I imagine that it is possible for a pair of nonisomorphic finite …

DISCLAIMER: My relationship with noncommutative algebraic geometry is that of a curious, ignorant bystander. I confess that I know very little about noncommutative algebraic geometry, but I am interes …

For a fixed prime $p$, the Sylow $p$-subgroups of a given finite group are all conjugate. Here are some more examples of situations in which we find that subgroups of a finite group defined by a certa …

Let $G$ be a finite group, let $F$ be a finite field and let $F[G]$ be the group algebra of $G$ over $F$. What is known about the structure of the group of units $F[G]^\times$? Of course, it must con …

This is problem 11.123 in the Kourovka notebook: For a given group $G$, define the following sequence of groups: $A_1(G) = G$, $A_{i+1}(G) = \operatorname{Aut}(A_i(G))$. Does there exist a finite gro …

(All groups in the following discussion are assumed to be finite.) Character induction is an operation that produces a character of a group given a character of a subgroup. I'm aware that there are ot …

Edit 20/12: I added a more precise question at the bottom of the post. Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by …

Let $G$ be a finite group. For each $x\in G$, the centralizer $\mathbf{C}_G(x)$ must contain $\langle x\rangle$. QUESTION: What are some interesting results of the following form: Given some bound on …

Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define: $W^G=\sum_ …

Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution! I asked this on math.stackexchange a couple of days ago, but it didn't attract any atten …

EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are …

This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups": “... the classification of finite simple groups is an exercise in taxonomy. This is obvious to the expert and t …

Have there been any attempts to extend the "F_un" analogy to the representation theory of finite groups? If I might be allowed some speculation: If combinatorics can be regarded as analagous to linear …